Waikiki and downtown Honolulu can realistically be done without a car. Waikiki itself is walkable, and public shuttles and buses connect it with the downtown area and the most important sites (Pearl Harbor, Polynesian Cultural Center, etc.)
If you plan to further your explorations on Oahu, rent a car. Reserve your vehicle in advance, particularly during peak travel times and during major special events. Most large rental car companies have airport counters and complimentary transportation for pickup/drop-off back at the airport upon departure.
If you are staying in Waikiki, consider renting a car only on the days you plan to use it. You don't need a car in Waikiki itself, and hotel parking garages charge everyone (even hotel guests) close to $20 a day, plus tips. Parking is another reason to consider taking TheBus or a shuttle to downtown Honolulu.